Understanding Scalars and Vectors
Vectors and Scalars: Key Concepts
A scalar has only magnitude; a vector has magnitude and direction.
In this course, vectors will be limited to two dimensions (no more than 2D) unless you’re studying higher-dimensional calculus.
Example intuition: describing 8 apples with no direction is not a vector description; a vector requires both how much and which way.
Distance vs Displacement
Distance: the total length of the path traveled. It accumulates as you move.
Example concept: walking from Starbucks to the library, then to another Starbucks, etc. The total distance accumulates as you go.
Displacement: the straight-line vector from your starting point to your final position. It has both magnitude and direction.
The magnitude of displacement is the straight-line distance from start to end, regardless of the path taken.
Important wording: displacement is about change of position; distance is about the amount travelled along the route.
In the example walk:
Distances along segments add to a total distance of 120 m (e.g., 20 m, 40 m, 60 m as highlighted in the transcript).
If you start and end at the same location (start UC and end back at UC in the example), the displacement magnitude is zero (a vector quantity with length 0).
Key relation:
The magnitude of displacement is always less than or equal to the total distance traveled:
|oldsymbol{ ext{displacement}}| \, \le \, ext{total distance}.
How to describe displacement direction in 1D: you may assign a positive direction (e.g., West) and a negative direction (e.g., East). In many one-dimensional contexts, explicit vector notation isn’t required, but you should still state the direction.
In 2D problems, use full vector notation and specify the direction (e.g., west, north, etc.).
Distance vs Displacement: Practical Notes
Reading questions carefully matters: magnitude vs displacement will be asked with care (e.g., if asked for the magnitude of displacement, give the length only; if asked for displacement, include both magnitude and direction).
Real-world relevance: pedometer-style thinking distinguishes how far you went (distance) from where you ended up (displacement).
One-Dimensional Motion (1D) and Kinematics
One-dimensional motion: motion along a single axis (e.g., up/down or along a straight road).
Example: a ball thrown straight up and falling back down; motion along the y-axis is 1D.
Key 1D quantities:
Distance (magnitude only, scalar) and displacement (vector along the line).
Speed is the scalar rate of distance change; velocity is the vector rate of displacement change.
Definitions from the lecture:
Distance traveled is the total path length; velocity requires direction.
Speed vs Velocity:
Speed = rate of change of distance (scalar).
Velocity = displacement over time (vector).
In 1D, velocity can often be described with a sign (positive/negative direction) along the axis.
Example discussion: one-dimensional kinematics for a drop or a car on a straight road uses both magnitude and direction to describe motion.
Average Speed vs Average Velocity
Average speed: the total distance traveled divided by the total time:
v_{ ext{avg}} = \frac{D}{\Delta t}
D is total distance traveled (scalar).
Example: a vehicle travels at 80 km/h for 0.5 h and 60 km/h for 1.5 h. Total distance = 80\times0.5 + 60\times1.5 = 40 + 90 = 130\,\text{km}, total time = 0.5 + 1.5 = 2.0\,\text{h}, so v_{ ext{avg}} = \frac{130}{2} = 65\,\text{km/h}.
Average velocity: the displacement divided by the total time; it is a vector:
\mathbf{v}_{\text{avg}} = \frac{\Delta \boldsymbol{r}}{\Delta t}.
If the net displacement is zero (you end where you started), the average velocity magnitude is zero, even though you may have traveled a nonzero distance.
Practical notes for writing:
When reporting average velocity, include direction (e.g., 0 m of displacement is typically written as 0 with no direction, but if displacement is nonzero, include the direction).
In quiz environments, partial credit is possible; be explicit about both magnitude and direction when needed.
One-dimensional displacement example:
If the finish line is 1/8 mile west of the start, the displacement is westward with magnitude 1/8 mile; the velocity would include the westward direction.
Instantaneous Speed and Velocity (Conceptual)
Instantaneous speed vs. average speed:
Instantaneous speed is the speed at a particular moment (the slope of the distance-versus-time curve at that moment is not directly used for distance, but the idea parallels velocity concepts).
The transcript discusses Hussein Bolt-like examples: a 100 m sprint, time ≈ 9.58 s, giving an average speed of about \frac{100}{9.58} \approx 10.4\,\text{m/s}.
Maximum average speed is not meaningful because average over an infinitesimal interval is not well-defined; speed/velocity are defined over intervals, with instantaneous values at a moment in time.
For velocity in a sprint, the average velocity over the race would be displacement (100 m) over total time; if you end where you started, average velocity is 0.
Graphical Interpretation: Displacement vs Time and Velocity
The slope of the displacement-versus-time graph corresponds to velocity:
If the graph is a straight line, velocity is constant; the slope is constant.
If velocity is constant, acceleration is zero (no change in velocity over time).
In contrast, a distance-versus-time graph does not directly give velocity because distance is not a vector; the slope of a distance-time graph is not a velocity in the vector sense.
Examples and Worked Scenarios (Key Takeaways)
Example: 8 apples with no direction — not a vector; needs magnitude and direction to be a vector.
Example path (distance and displacement):
Start at UC; go 20 m to bookstore; 40 m to Starbucks; 60 m to another location; return to UC (end point = start).
Total distance traveled: 120 m.
Displacement: zero (end = start); magnitude = 0 with direction not defined.
Imperial-unit example: finish line is 1/8 mile west of the start (emphasizes explicit direction in displacement).
Sign digits reminder: use significant digits appropriately; see course link for a dedicated tutorial on significant digits.
Velocity vs. speed in 1D practice:
Write velocity as a vector with a direction (e.g., west). If you don’t want to specify a direction in a particular context, you may use a scalar notation like v for speed, but for velocity always indicate direction when relevant.
Practical motion planning note:
When considering time-to-travel and reaction times (e.g., car distances and speeds), practical estimates (like “about 1.1 car lengths”) illustrate how speed and perception affect safe stopping distances.
Foundational Principles and Real-World Relevance
Distinguish path length (distance) from net position change (displacement): everyday motion involves both, and the distinction is crucial in physics problem solving.
Scalars vs vectors: many quantities you encounter in daily life are scalars (mass, temperature) while motion involves vectors (position, velocity, displacement).
The magnitude of displacement is the straight-line distance from start to end; the actual travel path can be longer due to detours.
In real-world contexts, you’ll encounter both 1D and 2D motion; this course frames vectors up to 2D for clarity and foundation before moving to higher dimensions.
Connections to Foundational Principles and Formulas
Core definitions:
Displacement: \Delta \boldsymbol{r} = \boldsymbol{r}{f} - \boldsymbol{r}{i} with magnitude |\Delta \boldsymbol{r}| and direction.
Distance: the total path length traveled (scalar).
Speed: v_{ ext{avg}} = \frac{D}{\Delta t} where D is total distance.
Velocity: \mathbf{v}_{\text{avg}} = \frac{\Delta \boldsymbol{r}}{\Delta t} with direction.
Relationship:
|\Delta \boldsymbol{r}| \le D with equality only when motion is a straight line along the displacement direction without detours.
Graphical interpretation:
Slope of the displacement-time graph gives velocity; a straight-line displacement-time graph indicates constant velocity; if velocity not changing, acceleration is zero.
Dimensional scope:
Vectors in this course are limited to two dimensions; higher-dimensional treatment is left for advanced topics.
Quick Reference: Key Formulas (LaTeX)
Average speed:
v_{ ext{avg}} = \frac{D}{\Delta t}Average velocity:
\mathbf{v}_{\text{avg}} = \frac{\Delta \boldsymbol{r}}{\Delta t}Displacement and its magnitude:
\Delta \boldsymbol{r} = \boldsymbol{r}{f} - \boldsymbol{r}{i} , \quad |\Delta \boldsymbol{r}|Distance vs displacement relation:
|\Delta \boldsymbol{r}| \le D